The measurement of prompt D-meson production as a function of multiplicity in p-Pb collisions at TeV with the ALICE detector at root s(NN) the LHC is reported. D-0, D+ and Dau+ mesons are reconstructed via their hadronic decay channels in the centre-of-mass rapidity range -0.96 <y(cms) <0.04 and transverse momentum interval 1 <p(T) <24 GeV/c. The multiplicity dependence of D-meson production is examined by either comparing yields in p-Pb collisions in different event classes, selected based on the multiplicity of produced particles or zero-degree energy, with those in pp collisions, scaled by the number of binary nucleon-nucleon collisions (nuclear modification factor); as well as by evaluating the per-event yields in p-Pb collisions in different multiplicity intervals normalised to the multiplicity-integrated ones (relative yields). The nuclear modification factors for D-0, D+ and D*(+) are consistent with one another. The D-meson nuclear modification factors as a function of the zero-degree energy are c